Abstract

The hypothesis that brain pathways form 2D sheet-like structures layered in 3D as "pages of a book" has been a topic of debate in the recent literature. This hypothesis was mainly supported by a qualitative evaluation of "path neighborhoods" reconstructed with diffusion MRI (dMRI) tractography. Notwithstanding the potentially important implications of the sheet structure hypothesis for our understanding of brain structure and development, it is still considered controversial by many for lack of quantitative analysis. A means to quantify sheet structure is therefore necessary to reliably investigate its occurrence in the brain. Previous work has proposed the Lie bracket as a quantitative indicator of sheet structure, which could be computed by reconstructing path neighborhoods from the peak orientations of dMRI orientation density functions. Robust estimation of the Lie bracket, however, is challenging due to high noise levels and missing peak orientations. We propose a novel method to estimate the Lie bracket that does not involve the reconstruction of path neighborhoods with tractography. This method requires the computation of derivatives of the fiber peak orientations, for which we adopt an approach called normalized convolution. With simulations and experimental data we show that the new approach is more robust with respect to missing peaks and noise. We also demonstrate that the method is able to quantify to what extent sheet structure is supported for dMRI data of different species, acquired with different scanners, diffusion weightings, dMRI sampling schemes, and spatial resolutions. The proposed method can also be used with directional data derived from other techniques than dMRI, which will facilitate further validation of the existence of sheet structure.

(a) To investigate the existence of sheet structure, we consider the relation of the Lie bracket [V, W](p) (black arrow) to the plane spanned by vectors V(p) (red arrow) W(p) and (blue arrow). More specifically, we evaluate the normal component of the Lie bracket [V, W]⊥(p) (green arrow). When it is zero, sheet structure exists. (b) Definition of the Lie bracket as the closure R(p) when trying to move around in an infinitesimal loop along the integral curves of V and W. Here, the loop corresponding to the closure R1is displayed ().

Mean absolute error (first row) and range (second row) of estimates for different voxel sizes δ = {0.5, 1,2} mm (a–c, columns). Each color plot shows the results for different choices for the kernel size N (indicated in voxel on the left side of each graph and in mm on the right side of each graph) and different settings for β. We used κ = 1/ρ = 1/26 mm−1, k = 250, and p = (10, −10,0).

Mean (asterisks) and range (error bars) of for different voxel sizes δ = {0.5, 1,2} mm (a–c, columns) and the two different implementations (rows). We used N = 11 voxel and β = 1, and hmax = 5 voxel and Δh = 0.5 voxel (the corresponding values in mm are given above each plot). Each plot shows the estimates in the case of sheet (green, indicated by the dashed line) and non-sheet (red, ) for different SNR levels k. We set κ = 1/ρ = 1/26 mm−1, and p = (10, −10,0).

Mean (asterisks) and range (error bars) of for different ‘spatial neighborhood settings’ (a–c, columns): N = {3,7,11} voxel with β = 1 with for the coordinate implementation (top row) and hmax = {1,3,5} with voxel with Δh = 0.5 voxel for the flows-and-limits implementation (the corresponding values in are noted above each plot). Each plot shows the estimates in the case of sheet (green) and non-sheet (red) for different dropout fractions. We set κ = 1/ρ = 1/26 mm−1, k = 250, and p = (10, −10,0).

Lie bracket and SPI computation on real in vivo HCP data: a comparison between the flows-and-limits and coordinate methods. (a) Standard deviation of the Lie bracket normal component estimates. Yellow arrows indicate areas where the mean is close to zero and the standard deviation is spatially uniform and low. The values in these areas were used to find an appropriate value for λ. (b) Sheet tensors on coronal, sagittal, and axial slices, for the flows-and-limits and coordinate implementations (λ = 0.008, sheet tensors with SPI < 0.2 are not shown for clarity). White arrow: crossing sheets found with the coordinate implementation, but not with the flows-and-limits method; blue arrow: crossing sheets found with the flows-and-limits method; red arrow: high SPI between the cingulum and the corpus callosum.